If $T$ is an invertible matrix, and for matrices $B$ and $C$, and we have that $∥TA−TB∥<ϵ$, can we say that A and B are close to each other in some sense?
Intuitively the above shows that $TA\approx TB$ and multiplying by $T^{−1}$ we have that $A\approx B$...
There must be some easy inequality to take advantage of here.
NOTE: I am not asking for an inequality independent of $T$, it may very well depend on $\|T\|$. But I would like something tight, if possible.
Not really. For any two matrices $A$ and $B$ and any $\epsilon>0$, put $T=\frac\epsilon{2\|A-B\|}I$. Then $\|TA-TB\|=\frac\epsilon2<\epsilon$. So you would be able to do this for any two matrices.
If you require this to be true for any $\epsilon>0$, then this would imply $TA=TB$ which of course implies $A=B$.
EDIT: The best bound we can do is $$\|A-B\|=\|T^{-1}(TA-TB)\|\leq\|T^{-1}\|\|A-B\|<\|T^{-1}\|\epsilon.$$